cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A206225 Numbers j such that the numbers Phi(j, m) are in sorted order for any integer m >= 2, where Phi(k, x) is the k-th cyclotomic polynomial.

Original entry on oeis.org

1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 15, 20, 24, 16, 30, 22, 11, 21, 26, 28, 36, 42, 13, 34, 40, 48, 32, 60, 17, 38, 54, 27, 19, 33, 44, 50, 25, 66, 46, 23, 35, 39, 52, 45, 56, 72, 90, 84, 78, 70, 58, 29, 62, 31, 51, 68, 80, 96, 64, 120
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Comments

Based on A002202 "Values taken by totient function phi(m)", A000010 can only take certain even numbers. So for the worst case, the largest Phi(k,m) with degree d (even positive integer) will be (1-k^(d+1))/(1-k) (or smaller) and the smallest Phi(k,m) with degree d+2 will be (1+k^(d+3))/(1+k) (or larger).
Note that (1+k^(d+3))/(1+k)-(1-k^(d+1))/(1-k) = (k/(k^2-1))*(2+k^d*(k^3-(k^2+k+1))) >= 0 since k^3 > k^2+k+1 when k >= 2.
This means that this sequence can be segmented into sets in which Phi(k,m) shares the same degree of polynomial and it can be generated in this way.

Examples

			For k such that A000010(k) = 1,
  Phi(1,m) = -1 + m,
  Phi(2,m) = 1 + m,
  Phi(1,m) < Phi(2,m),
  so, a(1)=1, a(2)=2.
For k > 2 such that A000010(k) = 2,
  Phi(3,m) = 1 + m + m^2,
  Phi(4,m) = 1 + m^2,
  Phi(6,m) = 1 - m + m^2.
For m > 1, Phi(6,m) < Phi(4,m) < Phi(3,m), so a(3)=6, a(4)=4, and a(5)=3 (noting that Phi(6,m) > Phi(2,m) when m > 2, and Phi(6,2) = Phi(2,2)).
For k such that A000010(k) = 4,
  Phi(5,m) = 1 + m + m^2 + m^3 + m^4,
  Phi(8,m) = 1 + m^4,
  Phi(10,m) = 1 - m + m^2 - m^3 + m^4,
  Phi(12,m) = 1 - m^2 + m^4.
For m > 1, Phi(10,m) < Phi(12,m) < Phi(8,m) < Phi(5,m), so a(6) = 10, a(7) = 12, a(8) = 8, and a(9) = 5 (noting Phi(10,m) - Phi(3,m) = m((m^2 + m + 2)(m - 2) + 2) >= 4 > 0 when m >= 2).

Links

Crossrefs

Cf. A194712, A000010, A002202, A032447.

Programs

  • Mathematica
    t = Select[Range[400], EulerPhi[#] <= 40 &]; SortBy[t, Cyclotomic[#, 2] &]

A206942 Numbers of the form Phi_k(m) with k > 2 and |m| > 1.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 17, 21, 26, 31, 37, 43, 50, 57, 61, 65, 73, 82, 91, 101, 111, 121, 122, 127, 133, 145, 151, 157, 170, 183, 197, 205, 211, 226, 241, 257, 273, 290, 307, 325, 331, 341, 343, 362, 381, 401, 421, 442, 463, 485, 507, 521, 530, 547, 553
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Comments

Phi_k(m) denotes the k-th cyclotomic polynomial evaluated at m.
We can see that for any integer b, b = Phi_2(b-1). However, if we make k>2 and |m|>1, Phi(k,m) are always positive integers that do not traverse the positive integer set.
The Mathematica program can generate this sequence to arbitrary upper bound maxdata without user's chosen of parameters. The parameter determination part of this program is explained in A206864.

Examples

			a(1) = 3 = Phi_6(2) = Cyclotomic(6,2).
a(2) = 5 = Phi_4(2) = Cyclotomic(4,2).
...
a(15) = 61 = Phi_5(-3) = Cyclotomic(5,-3).

Links

Crossrefs

Cf. A206225, A206710, A194712, A206292, A206864.
Cf. A006511 for phiinv function in the Mathematica program.

Programs

  • Julia
    using Nemo
function isA206942(n)
    if n < 3 return false end
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))        # Fouvry & Levesque & Waldschmidt
    for k in 3:K
        c = cyclotomic(k, x)
        for m in 2:M
            n == subst(c, m) && return true
        end
    end
    return false
end
L = [n for n in 1:553 if isA206942(n)]; print(L) # Peter Luschny, Feb 21 2018
  • Mathematica
    phiinv[n_, pl_] :=  Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 560; max =  Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb =  2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], EulerPhi[#] <= eb &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an =  SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, a = Append[a, cc]]; i = 2;  While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]
(* Alternatively: *)
isA206942[n_] := If[n < 3, Return[False],
    K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
    For[k = 3, k <= K, k++, For[x = 2, x <= M, x++,
        If[n == Cyclotomic[k, x], Return[True]]]];
    Return[False]
]; Select[Range[555], isA206942] (* Peter Luschny, Feb 21 2018 *)

A206292 Numbers k such that cyclotomic polynomial Phi(k,-m) < Phi(j,-m) for any j > k and m >= 2.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 18, 30, 42, 48, 60, 66, 70, 78, 90, 102, 120, 126, 150, 180, 210, 240, 270, 300, 330, 420, 450, 462, 480, 510, 540, 630, 660, 690, 780, 840, 870, 924, 1050, 1092, 1140, 1260, 1320, 1470, 1560, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Examples

			For k such that A000010(k) = 1:
  Phi(1, -m) = -1 - m,
  Phi(2, -m) = 1 - m,
  Phi(1, -m) <  Phi(2, -m),
  so a(1) = 1, a(2) = 2.
For k > 2 such that A000010(k) = 2:
  Phi(3, -m) = 1 - m + m^2,
  Phi(4, -m) = 1 + m^2,
  Phi(6, -m) = 1 + m + m^2.
When integer m > 1, Phi(3, -m) < Phi(4, -m) < Phi(6, -m), so a(3) = 3, a(4) = 4, and a(5) = 6.
For k > 6 such that A000010(k) = 4:
  Phi(8, -m) = 1 + m^4,
  Phi(10, -m) = 1 + m + m^2 + m^3 + m^4,
  Phi(12, -m) = 1 - m^2 + m^4.
When integer m > 1, Phi(12, -m) < Phi(8, -m) < Phi(10, -m), so a(6) = 12.

Links

Crossrefs

Cf. A194712, A206225, A000010, A002202, A032447.

Programs

  • Mathematica
    t = Select[Range[4000], EulerPhi[#] <= 1000 &]; t =  SortBy[t, Cyclotomic[#, -2] &]; DeleteDuplicates[Table[Max[Take[t, n]], {n, 1, Length[t]}]]

A206710 This irregular table contains indices j, k, l,... in each row such that the values Phi(j,-m) < Phi(k,-m)< Phi(l,-m)< ... of cyclotomic polynomials Phi(.,.) are sorted given any constant integer argument m >= 2.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 8, 10, 7, 9, 18, 14, 30, 20, 24, 16, 15, 11, 22, 42, 13, 28, 36, 21, 26, 17, 40, 48, 32, 60, 34, 19, 27, 54, 38, 66, 44, 25, 50, 33, 23, 46, 70, 78, 52, 90, 56, 72, 45, 84, 39, 35, 29, 58, 31, 62, 102, 68, 80, 96, 64, 120
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Comments

Based on A002202 "Values taken by totient function phi(m)", A000010 can only take certain even numbers. So for the worst case, the largest Phi(k,m) with degree d (even positive integer) will be (1-k^(d+1))/(1-k) (or smaller)and the smallest Phi(k,m) with degree d+2 will be (1+k^(d+3))/(1+k) (or larger).
(1+k^(d+3))/(1+k)-(1-k^(d+1))/(1-k)=(k/(k^2-1))*(2+k^d*(k^3-(k^2+k+1)))
k^3>k^2+k+1 when k>=2.
This means that this sequence can be segmented to sets in which Cyclotomic(k,m) shares the same degree of Polynomial and it can be generated in this way.

Examples

			For those k's that make A000010(k) = 1
Phi(1,-m) = -1-m
Phi(2,-m) = 1-m
Phi(1,-m) < Phi(2,-m)
So, a(1) = 1, a(2) = 2;
For those k's (k > 2) that make A000010(k) = 2
Phi(3,-m) = 1 - m + m^2
Phi(4,-m) = 1 + m^2
Phi(6,-m) = 1 + m + m^2
Obviously when integer m > 1, Phi(3,m) < Phi(4,m) < Phi(6,m)
So a(3)=3, a(4)=4, and a(5)=6
For those k's that make A000010(k) = 4
  Phi(5,-m) = 1 - m + m^2 - m^3 + m^4
  Phi(8,-m) = 1 + m^4
Phi(10,-m) = 1 + m + m^2 + m^3 + m^4
Phi(12,-m) = 1 - m^2 + m^4
Obviously when integer m > 1, Phi(5,m) < Phi(12,m) < Phi(8,m) < Phi(10,m),
So a(6) = 5, a(7) = 12, a(8) = 8, and a(9) = 10.
The table starts
1,2;
3,4,6;
5,12,8,10;

Crossrefs

Cf. A206292, A194712, A206225, A000010, A032447.

Programs

  • Mathematica
    t = Select[Range[400], EulerPhi[#] <= 40 &]; SortBy[t, Cyclotomic[#, -2] &]

A206944 Numbers Phi_k(m) with integer k > 2, |m| > 1 but k != 2^j (j > 1).

Original entry on oeis.org

3, 7, 11, 13, 21, 31, 43, 57, 61, 73, 91, 111, 121, 127, 133, 151, 157, 183, 205, 211, 241, 273, 307, 331, 341, 343, 381, 421, 463, 507, 521, 547, 553, 601, 651, 683, 703, 757, 781, 813, 871, 931, 993, 1057, 1093, 1111, 1123, 1191, 1261, 1333, 1407, 1483
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Comments

Phi_k(m) denotes the cyclotomic polynomial numbers Cyclotomic(k,m).
There is a property for Cyclotomic(k,m):
Cyclotomic(k^(j+1),m) = Cyclotomic(k,m^(k^j)).
So actually when k=2^(j+1), j is a positive integer,
Cyclotomic(k,m) = Cyclotomic(2,m^(2^j)) = 1+m^(2^j).
If these cases are excluded from A206942, this sequence is obtained.
This sequence is a subsequence of A206942.
Sequence A059054 is a subsequence of this sequence.
The Mathematica program can generate this sequence to arbitrary boundary maxdata without a user's choice of parameters. The parameter determination part of this program is explained at A206864.

Examples

			a(1) = 3 = Phi(6,2).
5 = Phi(4,2) = Phi(2,4) so excluded.
a(2) = 7 = Phi(3,2).

Crossrefs

Cf. A206942, A194712, A059054, A206864.

Programs

  • Mathematica
    phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 1500; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && ((! IntegerQ[Log[2, #]]) || (# <= 2))) &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an = SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata,
  a = Append[a, cc]]; i = 2; While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]

A206945 Prime numbers Phi(k,m) with integer k > 2, |m| > 1, and k != 2^j (j > 1).

Original entry on oeis.org

3, 7, 11, 13, 31, 43, 61, 73, 127, 151, 157, 211, 241, 307, 331, 421, 463, 521, 547, 601, 683, 757, 1093, 1123, 1483, 1723, 2551, 2731, 2801, 2971, 3307, 3541, 3907, 4423, 4561, 4831, 5113, 5419, 5701, 6007, 6163, 6481, 8011, 8191, 9091, 9901, 10303, 11131
Offset: 1

Views

Author

Lei Zhou, Feb 13 2012

Keywords

Comments

Phi(k,m) denotes the cyclotomic polynomial numbers Cyclotomic(k,m).
These are the prime terms of A206944.
A059055 is a subsequence of this sequence.
The Mathematica program can generate this sequence to arbitrary boundary maxdata without a user's choice of parameters. The parameter determination part of this program is explained at A206864.

Examples

			Just taking prime terms from A206944:
A206944(1)=3 is prime, so a(1)=3 ...

Crossrefs

Cf. A059055, A206944, A206942, A206864, A194712.

Programs

  • Mathematica
    phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 12000; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && ((! IntegerQ[Log[2, #]]) || (# <= 2))) &]; t = SortBy[t, Cyclotomic[#, 2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[t[[i]], m]; cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a]
Showing 1-6 of 6 results.

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